We consider the problem of computing the best-fitting ReLU with respect to square-loss on a training set when the examples have been drawn according to a spherical Gaussian distribution (the labels can be arbitrary). Let $\opt < 1$ be the population loss of the best-fitting ReLU. We prove: \begin{itemize} \item Finding a ReLU with square-loss $\opt + \epsilon$ is as hard as the problem of learning sparse parities with noise, widely thought to be computationally intractable. This is the first hardness result for learning a ReLU with respect to Gaussian marginals, and our results imply --{\em unconditionally}-- that gradient descent cannot converge to the global minimum in polynomial time.

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We consider the problem of computing the best-fitting ReLU with respect to square-loss on a training set when the examples have been drawn according to a spherical Gaussian distribution (the labels can be arbitrary). Let \opt 1 be the population loss of the best-fitting ReLU. We prove: \begin{itemize} \item Finding a ReLU with square-loss \opt \epsilon is as hard as the problem of learning sparse parities with noise, widely thought to be computationally intractable. This is the first hardness result for learning a ReLU with respect to Gaussian marginals, and our results imply --{\em unconditionally}-- that gradient descent cannot converge to the global minimum in polynomial time. The algorithm uses a novel reduction to noisy halfspace learning with respect to 0/1 loss.

We consider the problem of computing the best-fitting ReLU with respect to square-loss on a training set when the examples have been drawn according to a spherical Gaussian distribution (the labels can be arbitrary). Let $\opt 1$ be the population loss of the best-fitting ReLU. We prove: \begin{itemize} \item Finding a ReLU with square-loss $\opt \epsilon$ is as hard as the problem of learning sparse parities with noise, widely thought to be computationally intractable. This is the first hardness result for learning a ReLU with respect to Gaussian marginals, and our results imply --{\em unconditionally}-- that gradient descent cannot converge to the global minimum in polynomial time. The algorithm uses a novel reduction to noisy halfspace learning with respect to $0/1$ loss.

We consider the problem of computing the best-fitting ReLU with respect to square-loss on a training set when the examples have been drawn according to a spherical Gaussian distribution (the labels can be arbitrary). Let $\mathsf{opt} < 1$ be the population loss of the best-fitting ReLU. We prove: 1. Finding a ReLU with square-loss $\mathsf{opt} + \epsilon$ is as hard as the problem of learning sparse parities with noise, widely thought to be computationally intractable. This is the first hardness result for learning a ReLU with respect to Gaussian marginals, and our results imply -{\emph unconditionally}- that gradient descent cannot converge to the global minimum in polynomial time. 2. There exists an efficient approximation algorithm for finding the best-fitting ReLU that achieves error $O(\mathsf{opt}^{2/3})$. The algorithm uses a novel reduction to noisy halfspace learning with respect to $0/1$ loss. Prior work due to Soltanolkotabi [Sol17] showed that gradient descent can find the best-fitting ReLU with respect to Gaussian marginals, if the training set is exactly labeled by a ReLU.